
 we propose ELM (REF ), a technique that enforces both embedding and logit margins for long-tail learning,
leveraging insights from
metric [1]} and representation learning [2]};

we establish the benefits of enforcing embedding and logit margins, by showing that ELM encourages a better approximation to the Bayes-optimal classifier (§); and,

 we present experiments on synthetic and real-world datasets that confirm the value of ELM against existing methods (§),
and in particular demonstrate the import of enforcing margins in both logit and embedding space.

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